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Statement

Theorem 2 (p. 295). $f_3^i(n) \ge 2n^3/27 - cn^2$.

Notation. For distinct points $X_1,\ldots,X_n$ in $k$-dimensional Euclidean space $E_k$, the paper writes $f_k^i(n)$, $f_k^e(n)$, $f_k^c(n)$ and $f_k^s(n)$ for the largest possible number of isosceles triangles (congruent or not), of equilateral triangles, of pairwise congruent triangles and of pairwise similar triangles among them, the maximum taken over all choices of the $n$ points (pp. 291–292). Here $c$, $c_1$, $c_2$ are positive constants, not necessarily the same at each occurrence (p. 291).

Source. P. Erdős and G. B. Purdy, Some extremal problems in geometry, III, Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton, 1975), Congress. Numer. XIV, Utilitas Math., Winnipeg, 1975, pp. 291–308. The edition read is identified on the source card. Theorem 2 on p. 295, proof pp. 295–296.

Read depth. Claims checked: the statement was read clause by clause on the printed page.

Proof pointer

Construction (pp. 295–296): put $[2n/3]$ distinct points on the unit circle in the plane $x_3=0$ about the origin and the remaining $n-[2n/3]$ points on the axis through its center, at $(0,0,i)$. Every axis point is equidistant from all circle points, so each pair of circle points with each axis point spans an isosceles triangle, which gives $\tfrac12\bigl((2n/3)-1\bigr)\bigl((2n/3)-2\bigr)(n/3)\ge (2/27)n^3-cn^2$ of them as printed.

Bears on

No Erdős problem page of the corpus cites this result.